Methods and Apparatus for Improving Factor Risk Model Responsiveness

ABSTRACT

Construction of factor risk models that better predict the future volatility of returns of a portfolio of securities such as stocks, bonds, or the like is addressed. More specifically, improved factor-factor covariance estimation is made even when the covariances change rapidly over time. Methods and techniques for achieving better accuracy, responsiveness, and stability of factor risk models are addressed.

The present application is a continuation under 35 U.S.C. 120 of U.S.patent application Ser. No. 14/203,807 filed on Mar. 11, 2014, which isa divisional under 35 U.S.C. 120 of U.S. patent application Ser. No.13/503,698 filed on Apr. 24, 2012, now U.S. Pat. No. 8,700,516, whichclaims the benefit of PCT/US11/37231 filed on May 19, 2011 which claimsthe benefit of U.S. Provisional Application Ser. No. 61/435,439, filedJan. 24, 2011, all of which are incorporated by reference herein intheir entirety, for all purposes.

FIELD OF INVENTION

The present invention relates generally to the estimation of the risk,or active risk, of an investment portfolio using factor risk models.More particularly, it relates to improved computer based systems,methods and software for more accurate estimation of the risk or activerisk of an investment portfolio. The invention addresses techniquesallowing a factor risk model's risk estimates to be more accurate, morestable, and more responsive.

BACKGROUND OF THE INVENTION

Financial time series analysis often assumes that the statisticalproperties of equity returns do not vary over time. However, thestatistical properties of actual returns data from financial markets dovary over time. In particular, empirical evidence suggests thatvolatility or risk, the square root of the variance of returns, changeswith time. FIG. 1 shows a plot of both predicted risk 202 from a globalfactor risk model and one month forward looking realized risk 200 for abroad global benchmark portfolio. The one month, realized forwardlooking risk 200 changes over time. Even within relatively short timeintervals, such as a few weeks, the realized risk is not constant,exhibiting intermittent spikes of both modest and large magnitude aswell as fluctuating noticeably. During periods of market turmoil, suchas late 2008, volatility surges from a low value varying between 10% and20% annual volatility to over 70% annual volatility in a matter of oneor two months.

The challenge for commercial risk model vendors is to produce riskmodels that predict future volatility, or, in other words, accuratelypredicting the realized risk 200 shown in FIG. 1. The quality of riskmodel predictions can be measured with respect to at least threemetrics:

-   -   1. Prediction Accuracy. The difference between the realized and        predicted volatilities.    -   2. Stability. The risk model predictions should not exhibit the        smaller, transient changes observed in realized risk. In other        words, the risk predictions should be smoother than the realized        risk. Such smoothness ensures that portfolio rebalancing and        risk management decisions are not driven by market transients of        shorter duration than the investment holding horizon.    -   3. Responsiveness. When the overall level of market volatility        rises or falls, the predicted risk should respond similarly with        as little time lag in the response as possible.

Stability and responsiveness both bear on how changes in realized riskare tracked by risk model predictions. On the one hand, stabilityrequires that smaller, temporary changes in realized volatility shouldnot appear in the risk model predictions. On the other hand,responsiveness requires that larger, sustained changes in realizedvolatility should appear. Thus smaller changes are interpreted as noisethat should not affect investment decisions while the larger changes areinterpreted as meaningful changes that can and should affect investmentdecisions. The difference between smaller and larger changes ortemporary and sustained changes depends, of course, on the manner inwhich the risk model is used. A portfolio manager who trades every daymay consider a weeklong change in realized volatility a sustained changethat should be captured by a high quality risk model, while a portfoliomanager who invests over a time horizon of months may consider aweeklong change in realized volatility a temporary effect that should befiltered out of a high quality risk model. In both cases, the portfoliomanager wants to react to meaningful changes in market volatility thatcause material changes to his or her investment decisions whilesimultaneously avoiding any overreaction to temporary, noisy marketconditions that may lead to unnecessary trading. Stability seeks toensure that the risk model predictions are smooth over a sufficientlylong period of time, while responsiveness seeks to ensure that the riskmodel predictions change and respond to market changes in volatilityover a sufficiently short period of time.

In FIG. 1, risk model accuracy is measured by the difference between therealized risk 200 and the predicted risk 202. Stability is measured bythe fact that the predicted risk 202 is smoother than the realized risk200. Responsiveness is determined by how well the predicted risk 202tracks the realized risk 200 when the overall level of volatilitychanges.

In FIG. 1, the predicted risk 202 is reasonably accurate during theearly years of the decade and from 2006 to 2009. However, it is lessaccurate in reducing the predicted volatility from 2003 to 2006, whenmarket volatility drops to a historic low, remaining low for severalyears. In particular, the gap 201 between the predicted and realizedrisk in 2003, indicated by the arrows, is more than 5% throughout mostof 2003, and the gap 203 between the predicted and realized risk in2009, indicated by the arrows, is more than 10% throughout most of 2009.Approximately twenty-four months elapse starting from the beginning of2003 when market volatility falls before the predicted and realizedvolatilities are at the same level. Similarly, the predictionsthroughout 2009 are significantly higher than the realized volatility.More particularly, gaps 201 and 203 are larger than desirable.

There are several well known mathematical modeling techniques forestimating the risk of a portfolio of financial assets such assecurities and for deciding how to strategically invest a fixed amountof wealth given a large number of financial assets in which topotentially invest.

For example, mutual funds often estimate the active risk associated witha managed portfolio of securities, where the active risk is the riskassociated with portfolio allocations that differ from a benchmarkportfolio. Often, a mutual fund manager is given a “risk budget”, whichdefines the maximum allowable active risk that he or she can accept whenconstructing a managed portfolio. Active risk is also sometimes calledportfolio tracking error. Portfolio managers may also use numericalestimates of risk as a component of performance contribution,performance attribution, or return attribution, as well as, otherex-ante and ex-post portfolio analyses. See for example, R. Litterman,Modern Investment Management: An Equilibrium Approach, John Wiley andSons, Inc., Hoboken, N.J., 2003 (Litterman), which gives detaileddescriptions of how these analyses make use of numerical estimates ofrisk and which is incorporated by reference herein in its entirety.

Another use of numerically estimated risk is for optimal portfolioconstruction. One example of this is mean-variance portfoliooptimization as described by H. Markowitz, “Portfolio Selection”,Journal of Finance 7(1), pp. 77-91, 1952 which is incorporated byreference herein in its entirety. In mean-variance optimization, aportfolio is constructed that minimizes the risk of the portfolio whileachieving a minimum acceptable level of return. Alternatively, the levelof return is maximized subject to a maximum allowable portfolio risk.The family of portfolio solutions solving these optimization problemsfor different values of either minimum acceptable return or maximumallowable risk is said to form an “efficient frontier”, which is oftendepicted graphically on a plot of risk versus return. There arenumerous, well known, variations of mean-variance portfolio optimizationthat are used for portfolio construction. These variations includemethods based on utility functions, Sharpe ratio, and value at risk.

Suppose that there are N assets in an investment portfolio and theweight or fraction of the available wealth invested in each asset isgiven by the N-dimensional column vector w. These weights may be theactual fraction of wealth invested or, in the case of active risk, theymay represent the difference in weights between a managed portfolio anda benchmark portfolio as described by Litterman. The risk of thisportfolio is calculated, using standard matrix notation, as

V=w ^(T) Qw  (1)

where V is the portfolio variance, a scalar quantity, and Q is an N×Npositive semi-definite matrix whose elements are the variance orcovariance of the asset returns. Risk or volatility is given by thesquare root of V.

The individual elements of Q are the expected covariances of securityreturns and are difficult to estimate. For N assets, there are N(N+1)/2separate variances and covariances to be estimated. The number ofsecurities that may be part of a portfolio, N, is often over onethousand, which implies that over 500,000 values must be estimated. Riskmodels typically cover all the assets in the asset universe, not justthe assets with holdings in the portfolio, so N can be considerablylarger than the number of assets in a managed or benchmark portfolio.

To obtain reliable variance or covariance estimates based on historicalreturn data, the number of historical time periods used for estimationshould be of the same order of magnitude as the number of assets, N.Often, there may be insufficient historical time periods. For example,new companies and bankrupt companies have abbreviated historical pricedata and companies that undergo mergers or acquisitions have non-uniquehistorical price data. As a result, the covariances estimated fromhistorical data can lead to matrices that are numerically illconditioned. Such covariance estimates are of limited value.

Factor risk models were developed, in part, to overcome these shortcomings. See for example, R. C. Grinold, and R. N. Kahn, ActivePortfolio Management: A Quantitative Approach for Providing SuperiorReturns and Controlling Risk, Second Edition, McGraw-Hill, New York,2000, which is incorporated by reference herein it its entirety, andLitterman.

Factor risk models represent the expected variances and covariances ofsecurity returns using a set of M factors, where M is much less than N,that are derived using statistical, fundamental, or macro-economicinformation or a combination of any of such types of information. Givenexposures of the securities to the factors and the covariances of factorreturns, the covariances of security returns can be expressed as afunction of the factor exposures, the covariances of factor returns, anda remainder, called the specific risk of each security. Factor riskmodels typically have between 20 and 80 factors. Even with 80 factorsand 1000 securities, the total number of values that must be estimatedis just over 85,000, as opposed to over 500,000.

In a factor risk model, the covariance matrix Q is modelled as

Q=B ^(T) ΣB+Δ ²  (2)

where B is an N×M matrix of factor exposures, Σ is an M×M matrix offactor-factor covariances, and Δ² is a matrix of specific variances.Normally, Δ² is assumed to be diagonal.

The factor-factor covariance matrix Σ is typically estimated from a timeseries of historical factor returns, f_(t), for each of the M factors,while the specific variances are estimated from a time series ofhistorical specific returns.

Risk models used in quantitative portfolio management partly address theissues of stability and responsiveness when predicting time varyingvolatility by relying on an exponentially weighted covariance estimatorsince this estimator places greater emphasis on current observations,implicitly assuming that the most recent subset of return values oftenvary around a constant value. The returns can be asset returns, or theymay be factor returns or specific returns used for estimating a riskmodel covariance. Given a time series of T returns {r_(t), r_(t−1),r_(t−2), . . . , r_(t−T+1)}, we form the weighted returns series {{tildeover (r)}_(t)}

{{tilde over (r)} _(t)}={(w _(t) ,r _(t)),(w _(t−1) ,r _(t−1)), . . .,(w _(t−T+1) ,r _(t−T+1))}  (3)

w _(t−k)=2^(−k/H) ,k=0, . . . ,T−1  (4)

where H is the half-life parameter. The exponentially weightedcovariance estimator gives

E[var(r _(t+1))|_(t)]≡{circumflex over (σ)}_(t+1) ²=var({tilde over (r)}_(t))  (5)

This is frequently seen expressed in the RiskMetrics™ specification inwhich the half-life is reformulated as a decay factor λ. See, forexample, J. Longerstaey and M. Spencer, RiskMetrics™—Technical Document,Morgan Guaranty Trust Company, New York, 4th ed., 1996, which isincorporated by reference herein it its entirety. Equation (5) can berewritten as:

{circumflex over (σ)}_(t+1) ²=λσ_(t) ²⁺(1−λ)r _(t) ²  (6)

Ease and speed of computation, robustness, and parsimony have largelybeen responsible for the widespread adoption of exponentially weightedcovariance estimates in commercial risk models. Exponential weightinggenerally improves the accuracy of the risk model.

However, when realized risk changes rapidly, the risk predictions ofrisk models using exponentially weighted covariance estimates often lagrealized risk changes over considerable periods of time. In other words,exponential weighting does not always lead to the desired level ofresponsiveness in a risk model. This lag is shown in FIG. 1 during 2003by gap 201 and 2009 by gap 203, for example. The predicted risk in FIG.1 202 is computed from a risk model that uses exponential weighting witha 125-day half life for volatility estimation and a 250-day half-lifefor correlations. A larger half life is used for the correlationestimation in order to ensure a stable estimate.

One problem with exponentially weighted covariance estimates recognizedand addressed by the present invention is that large returns have adisproportionate effect on the covariance estimate even with exponentialweighting. These large returns can inflate risk estimates, and theyimpact risk estimates for very long times, resulting in lagged riskpredictions, especially when volatility falls from a high level, such asshown by gap 201 in 2003 and gap 203 in 2009.

In order to produce stable risk predictions, risk models typicallyrequire a long history of data for the covariance estimate. The longerthe data history, however, the more likely it is that the return historywill span a time period over which the volatility of the older returnsis at a substantially different level than the volatility of the recentreturns. Although the exponentially weighted covariance estimate willgive the older return data less weight than the most recent returns, theresulting volatility forecast may noticeably lag, in other words, not beas responsive to the realized volatility results as desired, if thevolatility of the older return data is substantially different than thevolatility of the more recent return data.

One approach to the problem of lagging risk model predictions is to useshorter data histories and/or aggressive decay factors in order toreduce the influence of the older data on the forecasts. However, if thedata history is too short or the decay factors too aggressive, thestability of the risk model predictions may be jeopardized.

Other methods besides more aggressive half-lives have been proposed toaddress the issue of non-stationarity of asset returns, factor returns,and specific returns. For example, generalized autoregressiveconditional heteroskedasticity (GARCH) models have been proposed. See,for example, Tim Bollerslev, “Generalized Autoregressive ConditionalHeteroskedasticity”, Journal of Econometrics, 31:307-327, 1986, which isincorporated by reference herein it its entirety. However, GARCH modelsnormally produce risk models that are too unstable for use in commercialrisk models.

SUMMARY OF THE INVENTION

The present invention recognizes that the lag in responding to rapidlychanging market volatility in 2003 and 2009 can be improved upon whencompared with the responsiveness to existing risk models. One aspect ofthe present invention is to provide a methodology for improving riskmodel responsiveness with minimal negative impact on both the risk modelaccuracy and stability. In some cases, as addressed further below, theaccuracy and stability may also be improved.

One goal of risk model prediction in accordance with the presentinvention, then, is to obtain a smooth curve of predicted risks thatclosely tracks the realized risk but that does not exhibit lags when theoverall level of volatility changes substantially. Exponential weightingalone does not solve the problem, nor does the use of shorter datahistories or more aggressive decay factors.

Another goal of the present invention is to improve responsiveness ofthe predicted risk without substantially increasing the change inforecast risk from one period to another.

Another aspect of the present invention is to improve risk modelresponsiveness over long periods of time.

Among its several aspects, the present invention addresses three things:(1), improving the responsiveness of the risk model; (2), maintainingthe same level of stability found in traditional risk models, wherestability is measured using the change in month-to-month predicted risk;and (3), maintaining accurate risk predictions over long periods of timeduring which the overall level of market volatility may or may notchange substantially.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates predicted versus realized risk for a broad globalportfolio from 2000 to 2010;

FIG. 2 shows a computer based system which may be suitably utilized toimplement the present invention;

FIG. 3 illustrates unscaled and scaled factor returns corresponding to aglobal, fundamental factor risk model being estimated in July 2010;

FIG. 4 illustrates predicted versus realized risk of a broad globalbenchmark portfolio from 2000-2010;

FIG. 5 illustrates risk comparison for the broad global benchmarkportfolio from 2000-2009;

FIG. 6 illustrates average month-to-month predicted risk change forthree different risk models;

FIG. 7 illustrates total risk of a global benchmark using the threevariants of Axioma's global fundamental factor risk model for three riskmodel variants;

FIG. 8 illustrates total risk of an Asian Pacific benchmark using thethree variants of Axioma's global fundamental factor risk model forthree risk model variants;

FIG. 9 illustrates total risk of a European benchmark using the threevariants of Axioma's global fundamental factor risk model for three riskmodel variants;

FIG. 10 illustrates total risk of a US benchmark using the threevariants of Axioma's global fundamental factor risk model for three riskmodel variants;

FIG. 11 illustrates bias statistics for 45 different portfolios from2000 to 2010; and

FIG. 12 illustrates the average forecast change for 45 differentportfolios for three risk model variants.

DETAILED DESCRIPTION

The present invention may be suitably implemented as a computer basedsystem, in computer software which is stored in a non-transitory mannerand which may suitably reside on computer readable media, such as solidstate storage devices, such as RAM, ROM, or the like, magnetic storagedevices such as a hard disk or floppy disk media, optical storagedevices, such as CD-ROM or the like, or as methods implemented by suchsystems and software.

FIG. 2 shows a block diagram of a computer system 100 which may besuitably used to implement the present invention. System 100 isimplemented as a computer 12 including one or more programmedprocessors, such as a personal computer, workstation, or server. Onelikely scenario is that the system of the invention will be implementedas a personal computer or workstation which connects to a server 28 orother computer through an Internet or other network connection 26. Inthis embodiment, both the computer 12 and server 28 run software thatwhen executed enables the user to input instructions and calculations onthe computer 12, send the input for conversion to output at the server28, and then display the output on a display, such as display 22, or isprinted out, using a printer, such as printer 24, connected to thecomputer 12. The output could also be sent electronically through theInternet connection 26. In another embodiment of the invention, theentire software is installed and runs on the computer 12, and theInternet connection 26 and server 28 are not needed. In still a furtherembodiment, the Internet connection is replaced with a local areanetwork. As shown in FIG. 2 and described in further detail below, thesystem 100 includes software that is run by the central processing unitof the computer 12. The computer 12 may suitably include a number ofstandard input and output devices, including a keyboard 14, a mouse 16,CD-ROM drive 18, disk drive 20, monitor 22, and printer 24. It will beappreciated, in light of the present description of the invention, thatthe present invention may be practiced in any of a number of differentcomputing environments without departing from the spirit of theinvention. For example, the system 100 may be implemented in a networkconfiguration with individual workstations connected to a server. Also,other input and output devices may be used, as desired. For example, aremote user could access the server with a desktop computer, a laptoputilizing the Internet or with a wireless handheld device such as anIPad™, IPhone™, IPod™, Blackberry™, Treo™, or the like.

One embodiment of the invention has been designed for use on astand-alone personal computer running in Windows (Microsoft XP, Vista,Windows 7). Another embodiment of the invention has been designed to runon a Linux-based server system.

According to one aspect of the invention, it is contemplated that thecomputer 12 will be operated by a user in an office, business, tradingfloor, classroom, or home setting.

As illustrated in FIG. 2, and as described in greater detail below, theinputs 30 may suitably include historical unadjusted returns of thefinancial assets to be included in a factor risk model; historicalunadjusted factor returns for the factors of a factor risk model; andhistorical, unadjusted specific returns of a factor risk model.

As further illustrated in FIG. 2, and as described in greater detailbelow, the system outputs 32 may suitably include adjusted historicalfactor returns; an improved factor-factor covariance matrix for thefactor risk model; and an improved factor risk model.

The output information may appear on a display screen of the monitor 22or may also be printed out at the printer 24. The output information mayalso be electronically sent to an intermediary for interpretation. Forexample, risk predictions for many portfolios can be aggregated formultiple portfolio or cross-portfolio risk management. Or,alternatively, trades based, in part, on the factor risk modelpredictions, may be sent to an electronic trading platform. Otherdevices and techniques may be used to provide outputs, as desired.

With this background in mind, we turn to a detailed discussion of theinvention and its context. The invention is herein referred to asDynamic Volatility Adjustment (DVA). DVA seeks to find a weightingscheme for historical returns that transforms them so that they moreclosely resemble a weakly stationary time-series. In the discussion thatfollows, algorithms may be suitably implemented as software stored inmemory and executed by a processor or processors in computer 12. Datamay be input by a user or retrieved from a database or other storage.Data entered by a user may be entered using a keyboard, mouse,touchscreen display or other data entry device or means. Output data maybe printed by a printer, displayed by a display, transmitted over thenetwork to another user or users, or otherwise output utilizing anoutput device or means. Equity returns r_(t) are weakly stationary whenthe first two moments of their distributions are stationary:

E[r _(t)]=μ  (7)

cov(r _(t) ,r _(t−τ))=γ_(τ)  (8)

for any τ where we assume that r_(t) and r_(1−τ) have finite and timeinvariant first and second moments, and that these values only depend onτ. When τ=0, equation (8) becomes variance, which is often used as ameasure of market volatility. Weak stationarity is a handy conditionbecause it allows inferences and predictions to be made about futurereturns. See, for example, Ruey Tsay, Analysis of Financial Time Series,John Wiley & Sons Inc., 2005, which is incorporated by reference hereinit its entirety.

DVA seeks a weighting scheme for a set of historical factor returns datathat transforms its second moment into a weakly stationary statistic.Specific returns are not modified by the DVA algorithm as they are toounstable. Let f be the observed time history of factor returns, and g bea weighting function to be determined. Weak stationarity of thevolatility of (g_(t), f_(t)) requires

cov[(g _(t) ,f _(t)),(g _(t−τ) ,f _(1−τ))]=γ_(τ)  (9)

For a finite set of T observed returns, {f₁, f₂, f₃, f₄, . . . ,f_(T))}, where f₁ is the oldest return and f_(T) is the most recentreturn, we want the series of weighted returns,{(g₁, f₁), (g₂, f₂) (g₃,f₃), (g₄,f₄), . . . , (g_(T),f_(T))}, tofluctuate with a relatively constant level of variance, computed as{circumflex over (σ)}²=var{(g₁,f₁), (g₂, f₂), (g₃,f₃), (g₄, f_(t)), . .. , (g_(T),f_(T))}.

There are many weighting functions g_(t) that will satisfy weakstationarity of the covariance. As originally formulated, DVA includesthe following steps.

First, assume that the T data points can be grouped into N overlappingsegments of length K, where T=(N+1) K/2, and one final data segment oflength K/2. Each segment except the earliest shares half of the pointsof the segment immediately before it, and each segment except the latestshares half the points of the segment immediately after it. The latestsegment is only half the length of the others, and is the ‘reference’segment, containing the most recent data. For example, with T=12, N=5,and K=4, we obtain the segments:

{S ₁ }={f ₁ ,f ₂ ,f ₃ ,f ₄}

{S ₂ }={f ₃ ,f ₄ ,f ₅ ,f ₆}

{S ₃ }={f ₅ ,f ₆ ,f ₇ ,f ₈}

{S ₄ }={f ₇ ,f ₈ ,f ₉ ,f ₁₀}

{S ₅ }={f ₉ ,f ₁₀ ,f ₁₁ ,f ₁₂}

{S ₆ }={f ₁₁ ,f ₁₂}  (10)

Each segment of historical data is denoted by {S_(n)}, n=1, . . . N+1,and is used to define a distinct volatility regime. The last segment,{S_(N+1)}, is referred to as the reference chuck or reference segment.In practice, the values for T, N, and K would be larger than for thissimple example. For example, T is often the entire factor returnhistory. For Axioma's US Equity model, there are daily returns goingback to Jan. 3, 1995, which is more than 4000 factor returns. For afundamental factor risk model, T normally corresponds to four years ofdata, making T=1000. For a statistical factor risk model, T normallycorresponds to one year of data, making T=250. K normally corresponds toabout 6 months or 125. Hence, N=7 for a fundamental factor risk modeland N=3 for a statistical factor risk model. Compute the N mean absolutedeviations for each {S_(n)}:

$\begin{matrix}{{v_{n} = {\frac{1}{K}{\sum\limits_{i \in S_{n}}{f_{i}}}}},{n = 1},{{\ldots \mspace{14mu} 1};{v_{N + 1} = {\frac{1}{K/2}{\sum\limits_{i \in S_{N + 2}}{f_{i}}}}}}} & (11)\end{matrix}$

Second, compute the N scaling factors for each v_(n)

$\begin{matrix}{{\delta_{n} = \frac{v_{N + 1}}{v_{n}}},{n = 1},\ldots \mspace{14mu},{N + 1.}} & (12)\end{matrix}$

The N+1 scaling factors are clipped to lie within 0.8≤δ_(n1)≤1.25. Thisprevents the scaling values having too large an impact on the returns,which improves stability. This potentially adversely affects thestationarity of the resulting time series, but is imposed for the sakeof model stability and robustness to noisy data.

Third, assume a piecewise constant approximation of the N+1 scalingfactors δ_(n) to compute the T weighting values g_(t). That is

g _(T) =g _(T−1) = . . . =g _(T−K/2+1)=δ_(N+1)≡1

g _(T−K/2) =g _(T−K/2−1) = . . . =g _(T−K+1)==δ_(N)

g _(T−K) =g _(T−K−1) . . . =g _(T−3K/2+1)=δ_(N−1)  (13)

Since δ_(N+1)≡1, we have g_(T)=1 so that the most recently observedreturn is unchanged when weighted.

FIG. 3 shows both unscaled factor returns, f_(t), 204 as individualpoints and scaled returns, g_(t) f_(t), 206 drawn as a thin line for atime series of returns for a global fundamental factor risk model fromOctober 2006 to July 2010. The weighting scheme shown is for July 2010.FIG. 3 shows that, for July 2010, the historical factor returns areadjusted to slightly smaller values during most of 2008 in order forthose returns to have the same level of volatility as in July 2010. Thisis clear from the fact that several of the unscaled factor returnspoints 204 are much greater than or less than the scaled factor returns206.

The advantages of DVA can be seen in the results shown in FIGS. 4, 5,and 6. FIG. 4 revisits FIG. 1 and compares the predicted volatility of abroad global benchmark for the same period, with and without DVA, to thebenchmark's realized volatility. FIG. 4 has three lines: the realizedrisk 208, the predicted risk without DVA 210, and the predicted riskwith DVA 212.

In FIG. 4, the overestimation of risk is substantially reduced over 2003and in 2009 by incorporating DVA. When volatility stays at a constantlevel for several years, as occurs in 2005-2007, the predictions bothwith and without DVA converge, as overall levels of volatility becomestable for the duration.

FIG. 5 compares a DVA-enabled model with a non-DVA model that usesshorter, more aggressive half-lives. In the shortened half-life model,the half-life for volatility is changed from 125 days to 60 days, whilethe half-life for correlation is changed from 250 days to 125 days. FIG.5 shows three lines: realized risk 214, predicted risk without DVA butwith a shorter half life 216, and predicted risk with DVA but thestandard half life 218.

FIG. 5 shows that DVA yields similar responsiveness, when necessary, tothe model with a shorter half-life.

The present invention recognizes, for example, that in FIG. 5 that theDVA 218 and shorter half-life 216 predictions cross each other in mid2009. In the first half of 2009, the DVA predictions are more accurate,while in the second half of 2009, the shorter half life predictions aremore accurate. Thus, the present invention recognizes that DVA asoriginally proposed could be improved upon particularly with respect tolong term accuracy. This is one aspect that the present inventionaddresses.

Although both the DVA and shorter half life models have comparableresponsiveness, the stability of the DVA model is superior to that ofthe shorter half-life model. FIG. 6 compares the average change inpredicted risk from one month to the next for three different riskmodels over four time periods: 2000-2002; 2003-2005; 2006-2009; and2000-2009. The three risk models are the original risk model without DVA224; the risk model with DVA 222; and the risk model with shorter halflife 220. The relative change in risk model prediction gives aquantitative measure of the stability of the model. The most stable riskmodel is the original model, which is also the least responsive. Theleast stable risk model is the risk model with the shorter half life.The risk model with DVA is as responsive as the shorter half life model,but it has superior stability over three out of four time periods, andis identical for the other one. Hence, the DVA model represents a bettertrade-off between stability and responsiveness than either of the otherrisk models.

The present invention addresses a formulation of DVA that improves uponseveral aspects of the original formulation.

In the original DVA formulation, the weighting values g_(t) are constantover finite time intervals. The jump that occurs when the weightingvalues change from one scaling factor, δ_(n), to another can lead toundesirable changes in the risk model prediction. In other words, thefact that the weighting values approximate a non-differentiable functioncan negatively impact the stability of the risk predictions.

In the original formulation of DVA, the length of the reference chunk{S_(N+2)} is half that of the other chunks. This makes it difficult todefine the reference time horizon of the resulting risk model as thescaling values are defined over data segments of different length.

In the original DVA formulation, scaling values lie within [0.8, 1.25].Although this ensures a degree of stability, it also means that an olderperiod of excessively high volatility will never be scaled down by afactor of greater than 0.8, no matter how distantly in the past it lies.

In the original DVA formulation, the long term accuracy of the DVAenable risk model may be worse than that of a risk model with a shorterhalf life, as shown in FIG. 5 over the second half of 2009.

To address these issues, an improved version of DVA incorporates thefollowing improvements. This improved DVA is the preferred embodiment ofthe invention.

First, rather than segmenting the history of factor returns into Nsegments of length K and a final segment of length K/2, the history issegmented into only N segments of length K. Hence, with T=12, N=5, andK=4, we obtain the segments:

{S ₁ }={f ₁ ,f ₂ ,f ₃ ,f ₄}

{S ₂ }={f ₃ ,f ₄ ,f ₅ ,f ₆}

{S ₃ }={f ₅ ,f ₆ ,f ₇ ,f ₈}

{S ₄ }={f ₇ ,f ₈ ,f ₉ ,f ₁₀}

{S ₅ }={f ₉ ,f ₁₀ ,f ₁₁ ,f ₁₂}  (14)

With this change, the N scaling factors are redefined as

$\begin{matrix}{{\delta_{n} = \frac{v_{N}}{v_{n}}},{n = 1},\ldots \mspace{14mu},{N.}} & (15)\end{matrix}$

where now δ_(N)=1.

Secondly, rather than use a piecewise constant approximation to estimatethe weighting values, use cubic spline interpolation on the N scalingfactors δ_(n) to compute the T weighting values g_(t) assuming

g _(T−(n−1)K)=δ_(N−(n−1)) ,n=1, . . . ,N.  (16)

Since δ_(N)=1, we have g_(T)=1 so that the most recently observed returnis unchanged when weighted. Unlike the piecewise constant approximationof the original DVA formulation, this approximation varies smoothly andcontinuously.

Thirdly, rather than clip the scaling values to be within [0.8, 1.25],the requirement that the ratio between any two consecutive scalingfactors is no more than 10% is imposed. That is,

$\begin{matrix}{{0.9 \leq \frac{\delta_{n}}{\delta_{n - 1}} \leq 1.1},{n = 2},\ldots \mspace{14mu},{N.}} & (17)\end{matrix}$

This ensures stability while simultaneously allowing older time periodsof excessively high volatility to be appropriately scaled. It alsoimproves the cubic spline interpolation. If a series of scaling factorsis clipped at the same level, then the cubic-spline interpolationoscillates around the clipped value. The 10% figure was chosenempirically to balance responsiveness and stability, and to minimizeprediction differences between improved DVA and original DVA duringperiods of relatively stable volatility. These changes substantiallyimprove the performance of DVA.

FIGS. 7 to 10 compare three variants of risk model predictions torealized, 22-day (one month) volatility for four different benchmarks.The three risk model prediction variants are original DVA, improved DVA,and a risk model with a shorter half life (60 days for volatilities, 125days for correlations).

FIG. 7 shows the total realized risk 224 of a global benchmark comparedto predictions from three variants of Axioma's global fundamental factorrisk model: original DVA 230, the shorter half-life model 226, and theimproved DVA 228. The differences between model variants can be seenmost clearly from 2008 onwards. The short-term model 226 is the mostresponsive, as it should be since it has the most aggressive half-life.It overshoots the volatility peak in December 2008 substantially morethan the other two models, and then it falls the fastest (from thehighest value) throughout 2009. Original DVA 230 is more responsive inearly 2009 in that it drops more rapidly during this period than theother variants. However, over the rest of 2009, its predictions actuallytrend away from market volatility. That is, the accuracy of the originalDVA model in 2009 erodes in comparison with the other models. Theimproved DVA formulation 228 achieves an advantageous trade off inresponsiveness and accuracy. Unlike the shorter half-life model, it doesnot overshoot realized volatility as much in early 2009, and unlike theoriginal DVA, it tracks realized risk more accurately in late 2009.Similar results are obtained for other benchmarks.

FIG. 8 shows total risk results for an Asian-Pacific benchmarkportfolio. There are four lines: realized risk 232; the shorter halflife risk prediction 234; the original DVA risk prediction 238; and theimproved DVA prediction 236.

FIG. 9 shows total risk results for a European benchmark. There are fourlines: realized risk 240; the shorter half life risk prediction 242; theoriginal DVA risk prediction 246; and the improved DVA prediction 244.

FIG. 10 shows results for a US benchmark. There are four lines: realizedrisk 248; the shorter half life risk prediction 250; the original DVArisk prediction 254; and the improved DVA prediction 252.

In all cases, the original version of DVA is more responsive in January2009, but its accuracy erodes over the rest of 2009 in comparison toimproved DVA and the shorter half-life model.

FIG. 11 shows the average bias statistic for 45 different portfolios forthe three model variants. The bias statistic is taken over the time from2000 to 2010 on a monthly basis. For an unbiased risk model, the biasstatistic will be close to one. For each of the 45 portfolios, threebars are shown: a light bar on the left 260 representing the shorterhalf life risk model predictions; a medium bar on the right 264representing the original DVA predictions; and a dark bar in the center262 representing the improved DVA predictions. Over the broad range ofportfolios shown in FIG. 11, there is no significant difference in thebias statistics of the three different model variants. That is, for eachfactor, the three bars—original DVA 264, improved DVA 262 and theshorter half-life model 260—are essentially the same.

However, there are significant differences in the stability of the threemodel variants. FIG. 12 shows the forecast change statistic for 45different portfolios Forecast change gives a quantitative measure of theturnover of the risk model predictions, which is closely related to thestability of the risk model.

For each of the 45 portfolios, three bars are shown: a light bar on theleft 270 representing the shorter half life risk model predictions; amedium bar on the right 272 representing the original DVA predictions;and a dark bar in the center 274 representing the improved DVApredictions.

The improved DVA model predictions 274 show a clear reduction inforecast change in comparison with both the original DVA model 272 andthe shorter half-life model 270. In other words, its forecasts are muchsmoother, day on day, without losing accuracy. These results show thatthe improved DVA gives better responsiveness without sacrificingsmoothness of forecast.

While the present invention has been disclosed in the context of variousaspects of presently preferred embodiments, it will be recognized thatthe invention may be suitably applied to other environments consistentwith the claims which follow.

What is claimed is:
 1. A computer-implemented method for interactivelycomparing different portfolio risk estimates computed using a factorrisk model within a display interface, the method comprising:electronically receiving by a programmed computer a set of historicaltimes selected utilizing an input device interacting with the displayinterface; electronically receiving by the programmed computer aninvestment portfolio to be analyzed at each historical time selected;electronically receiving by the programmed computer a time serieshistory of factor returns at each historical time selected;electronically receiving by the programmed computer a time serieshistory of original factor risk models at each historical time selected,the original factor risk models comprising an exposure matrix, factorcovariance matrix, and specific variance matrix for each historical timeselected; displaying within a window of the display interface a firstgraphical representation of a time series history of the portfolio riskcomputed using the original factor risk models; displaying within thewindow of the display interface an alternative second graphicalrepresentation of a time series history of the portfolio risk, saidalternative representation computed by: calculating a set ofexponentially decaying weights with a fixed half-life corresponding tothe time series history of factor returns; computing a metric ofvolatility for each historical time; calculating a set of volatilityadjustment multipliers that is the ratio of most recent volatilitymetric to the measured volatility metric; determining that at least onevolatility adjustment multiplier is outside a predetermined range;adjusting the at least one volatility adjustment multiplier to a valuein the predetermined range; computing an alternative factor variance forthe time series of factor returns using the set of exponentiallydecaying weights, and volatility adjustment multipliers within the rangeand any adjusted volatility adjustment multiplier for any volatilitymultiplier determined to be outside the range; creating a modifiedfactor risk model by substituting the alternative factor variance ateach historical time into the original factor risk model; recomputingthe portfolio risk at each historical time using the modified factorrisk model; outputting a second graphical representation of a timeseries history of the portfolio risk computed using the modified factorrisk models on which the alternative second graphical representation isbased; and automatically identifying within the display interface timeperiods at which a risk difference in the portfolio risk in the firstgraphical representation and the second graphical representation islarger than a predetermined value.
 2. The computer-implemented method ofclaim 1 wherein the second graphical representation is displayed only ifthe risk difference is larger than the predetermined value.
 3. Thecomputer-implemented method of claim 1 further comprising: displayingwithin the window of the display interface a third graphicalrepresentation of realized risk.
 4. The computer-implemented method ofclaim 1 further comprising: aggregating risk predictions for manyportfolios for multiple portfolio management.
 5. Thecomputer-implemented method of claim 1 wherein 0.8 is less than or equalto the ratio of most recent volatility to the measured volatility metricwhich is less than or equal to 1.25.